 reserve m,n for Nat;
 reserve i,j for Integer;
 reserve S for non empty multMagma;
 reserve r,r1,r2,s,s1,s2,t for Element of S;
 reserve G for Group-like non empty multMagma;
 reserve e,h for Element of G;
 reserve G for Group;
 reserve f,g,h for Element of G;

theorem Th13:
  f * h = g iff f = g * h"
proof
  g * h" * h = g * (h" * h) by Def3
    .= g * 1_G by Def5
    .= g by Def4;
  hence f * h = g implies f = g * h" by Th6;
  assume f = g * h";
  hence f * h = g * (h" * h) by Def3
    .= g * 1_G by Def5
    .= g by Def4;
end;
